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Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103
Contents lists available at ScienceDirect
Journal of Non-Newtonian Fluid Mechanics
journal homepage: www.elsevier.com/locate/jnnfm
Isodense displacement flow of viscoplastic fluids along a pipe
G.V.L. Moisés a , M.F. Naccache
b , ∗, K. Alba
c , I.A. Frigaard
d
a Petrobras, Rua República do Chile 330, Rio de Janeiro, RJ, Brazil, 20031-170, Brazil b Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, Rio de Janeiro, RJ, Brazil,
22453-900, Brazil c Department of Engineering Technology, University of Houston, 4800 Calhoun Rd. Houston, TX, US, 77004, US d Department of Mathematics and Department of Mechanical Engineering, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada,
V6T 1Z2, Canada
a r t i c l e i n f o
Article history:
Received 30 May 2016
Revised 4 August 2016
Accepted 6 August 2016
Available online 26 August 2016
Keywords:
Viscoplastic fluids
Yield stress
Displacement flow
Carbopol
a b s t r a c t
We present the results of an experimental study of isodense displacement of a yield stress viscoplastic
fluid by a miscible Newtonian one in a horizontal pipe. A central type displacement flow develops for the
configuration given. Three distinct flow types belonging to this central displacement are identified namely
corrugated, wavy and smooth depending on the level of the residual layer variation along the pipe. The
transition between these flow regimes is found to be a function of the Reynolds number defined as the
ratio of the inertial stress to the characteristic viscous stress of the viscoplastic fluid.
Parameter ranges used in the current experimental study.
Parameter Range
β( °) 90 ˆ V 0 (mm/s ) 5 – 185 ˆ ˙ γc 2 – 100
ˆ τy (Pa ) 1.3 – 26.0
Re 0.02 −8 . 20
B 0.16 – 1.51
M 256 – 4350
Re N 87 – 3764
B N 360 – 48 ,200
At 0
Fr ∞
a
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3
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c
d
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(
Note that the Atwood number, defined as At = ( ρ1 − ˆ ρ2 ) / ( ρ1 +ˆ ρ2 ) , is zero in our study ( At = 0 ) since ˆ ρ1 = ˆ ρ2 = ˆ ρ . It is worth
mentioning that at times it is convenient to consider a Newtonian
Reynolds number, given by
Re N =
ˆ ρ ˆ V 0 D
ˆ μ= M(1 + B ) Re, (7)
and also a modified Bingham number, using a Newtonian viscous
stress:
B N =
ˆ τy D
8 μ ˆ V 0
= MB. (8)
Taghavi et al. [8] and Alba et al. [9] identified two regimes of
non-isodense yield stress displacements: central and slump. The
central regime is characterized by the central flow of the heavy
displacing fluid flowing within the light displaced layer, while in
the slump regime the displacing layer moves closer to the lower
region of the tube, underneath the displaced fluid. The transition
between these two types of displacement is governed by the ratio
of Re N / Fr
Re N F r
=
ˆ ρ1
([ ρ1 − ˆ ρ2 ] g D
3 )1 / 2
[ ρ1 + ˆ ρ2 ] 1 / 2
ˆ μ, (9)
where ˆ g is the gravitational acceleration, and Fr represents the
densimetric Froude number F r =
ˆ V 0 / (At g D
)1 / 2 . According to [8,9] ,
the transition between the central and slump flows occurs in the
range 600 < Re N / Fr < 800. In our experiments, since no density
different is applied ( At = 0 ), Re N / Fr tends to zero as Fr goes to in-
finity. Therefore, all our isodense experiments can be classified in
the central regime.
The parameter range covered in our experiments is given in
Table 2 . The characteristic shear rate range for all experiments is
2–100 1/ s , so the fitted Herschel Bulkley model quite well repre-
sents the rheological Carbopol behaviour during the fluid displace-
ment experiments. Besides, the complexities of low shear rate rhe-
ology are largely avoided. Every experiment can be represented by
set of dimensionless parameters ( Re, B, M ) or by dimensional ex-
erimental numbers ( Table 1 ), and the mean velocity ( V 0 ).
. Experimental results
.1. Typical experimental analysis
In this section we present in detail the analysis of a typical
isplacement experiment (taken from set A, for ˆ V 0 = 38 . 4 mm/s,
.e. Re = 1 . 66 , B = 0 . 22 and M = 400 ). After setting the tank pres-
ure and/or the choke position at the end of the flow loop, the gate
alve is opened; see Fig. 3 a for snapshots of the flow. The signals
f the upstream pressure and flow rate over time are presented in
ig. 4 . In this case, the tank pressure is initially set to 70.3 kPa. Af-
er opening the gate valve at ˆ t = 0 s, the upstream pressure drops
o a stable value of 48.3 kPa. Note that as flow develops over time,
he pressure and flow rate approach a mean value indicated by the
ashed lines in the figure. Once the experiment is over and the
hoke or gate valve is closed ( t ≈ 66 s ) the flow rate and pressure
aturally converge to their values prior to the experiment.
Fig. 3 .a presents the post processed images of the same exper-
ment as in Fig. 4 . The concentration contour for different time
tamps shows a finger of the brighter displacing fluid moving
hrough the dark displaced fluid from the left to the right. After
he front passes, it is possible to identify dark regions with lower
oncentrations at the top and bottom of the pipe and intermediate
right regions in the middle, suggesting the presence of a uniform
esidual layer on the walls. Since the concentration appears to re-
ain constant after the front passage, it seems that the front it-
elf plays an important role in the formation of the residual layer.
n Fig. 3 a it is also possible to identify detached Carbopol pieces
oving inside fluid 1 that are represented by the squiggly line in
ig. 3 b.
If the mean concentration ( C ) in a particular section of the
ow is constant, it is possible to estimate the residual thickness
ˆ r = ( D − ˆ d C ) / 2 , where ˆ D is the tube diameter and
ˆ d C =
√
C D is
he displacing fluid diameter obtained using a mass balance. As
escribed in [27,30] , the position of the displacing front in a spa-
iotemporal plot ( Fig. 3 b) is defined by the boundary between the
ark and light regions, with its slope being inversely proportional
o the front velocity. It is possible, using a mass balance, to calcu-
ate another equivalent diameter of the front by ˆ d f = √
ˆ V 0 / V f ˆ D , where
ˆ 0 and
ˆ V f are the mean and the front velocities, respectively. The
onstruction of ˆ d C can be considered as localized in space: depen-
ent on which C is used, whereas the construction of ˆ d f may be
onsidered to depend on the front velocity at a particular time.
If the displacement proceeds relatively steadily in time and
pace, we may justifiably compare ˆ d f and
ˆ d C . As an example,
he spatiotemporal plot in Fig. 3 b provides the front velocity
V f = 56 . 7 mm/s ) and the mean concentration ( C = 0 . 54 ) after the
G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103 95
Fig. 3. Displacement flow experiment carried for set A for ˆ V 0 = 38 . 4 mm/s ( Re = 1 . 66 , B = 0 . 22 and M = 400 ): a) Snapshots of the flow at ˆ t = 53 . 0 , 55 . 7 , 58 . 3 , 61 . 0 , 63 . 7 ,
and 66.3 s after opening the gate valve. The flow is from left to right as indicated by the arrow. The field of view is 600 mm by 19 mm, taken 2400 mm after the gate
valve. The bottom image is a colourbar of the concentration C . b) Spatiotemporal diagram of the same displacement showing the change in depth-averaged concentration C y values with streamwise distance, ˆ x , and time, t .
Fig. 4. Variation of the signals of the upstream pressure (top curve) close to the
gate valve ( x = 0 ) and the mean flow velocity (bottom curve) over time, for a typical
experiment carried using the experimental set A for ˆ V 0 = 38 . 4 mm/s . The broken
lines show the mean values of the developed flow. The experiment starts at t = 0 s
and finishes at t ≈ 66 s.
f
0
0
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t
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A
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ront passage. For this displacement, we have ˆ d f = 0 . 82 D and
ˆ d C = . 73 D . Note that the corresponding residual layer thickness to ˆ d C is
.27 radius of the pipe, which is close to those reported by [11] for
sodense displacement flows in a vertical pipe. In the case of buoy-
nt displacement flows, Alba and Frigaard [12] showed that bulk
easurements of the two diameters ˆ d C and
ˆ d f were relatively close
o each other over the range of experiments.
The dimensionless velocity contour, U =
ˆ U / V 0 , obtained by the
DV probe, is presented in Fig. 5 a as a function of time, ˆ t , and di-
ensionless radial position, r = 2 r / D . It is possible to identify the
ymmetry with respect to the tube center line. Due to the nature
f the UDV measurements there is always some refraction errors
lose to the lower wall of the pipe. To provide better understand-
ng of the UDV data, the corresponding data points to the lower
all were suppressed, and a model prediction is instead added.
nother way to interpret the UDV data is to study the stream-
ise velocity as a function of time at an averaged radial position,
s shown in Fig. 5 b. The UDV probe is fixed at ˆ x = 1560 mm and,
s a static observer, it measures the velocity contour of the fluid at
his particular location at different time stamps. Based on the data
n Fig. 5 , it is possible to identify 5 different stage in the local flow
described below) as the displacement front passes from upstream
o downstream. These stages can each be important in processes
elated to the fluid displacement, specially in the context of the
ipeline restart and shut-down:
I. Transient start-up: As the gate valve opens, displaced fluid 2
starts to move until the experiment steady state velocity ( V 0 ) is
reached.
II. Displaced fluid 2 steady state flow: Steady state flow of fluid 2
in a pipe of diameter ˆ D , which shows a typical plug-type profile
for viscoplastic fluids [12] (see also Fig. 6 ).
II. Displacement front region: We observe a period of time when
the measured velocity profile is influenced by the front region,
that contains both fluids 1 and 2.
V. Displacing fluid 1 steady state flow: Steady state flow of fluid 1
in a pipe of effectively reduced diameter ( d C =
ˆ D − ˆ h r ), due to
the formation of the residual layer at the wall. For example in
this stage, Fig. 5 a shows regions of zero velocity near the walls
and Fig. 5 b shows an increase in the streamwise velocity, due
to this diameter restriction.
V. Transient shut-down: As gate valve closes at the end of the ex-
periment, the pressure drop goes to zero and the flow ceases.
Fig. 6 shows the average streamwise velocity profile measured
or regions II and IV of the same experiment shown in Fig. 5 . Also
arked on Fig. 6 are the analytical solutions for fully developed
ow, based on the rheological parameters and the mean velocity.
n the case of region IV, we have plotted analytical solutions for
oth reduced diameters, ˆ d C and
ˆ d f ; in each case the displaced fluid
all layers are stationary. Note that there is slight skewness in
he velocity profiles with respect to the tube center evident from
ig. 6 which is commonly associated with the UDV measurements
8,9,12] .
.2. Residual layer thickness variation
In each experimental set, we observe variations in the resid-
al layer thickness along the tube. Fig. 7 a shows snapshots of
he concentration field after the front passes and Fig. 7 b shows
heir respective depth-averaged profiles. As the imposed velocity
ecreases, the amplitude of the residual layer variation increases
hich is better quantified in Fig. 7 b. Note that the interface mod-
lations shown in Fig. 7 a are similar to those reported by Swain
t al. [26] for light-heavy displacement flow of a viscoplastic fluid
96 G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103
Fig. 5. Data from Ultrasonic Doppler Velocimeter from experimental batch A for ˆ V 0 = 38 . 4 mm/s ( Re = 1 . 66 , B = 0 . 22 and M = 400 ): a) Contours of the dimensionless
streamwise velocity profiles with scaled radial distance, r = 2 r / D , and time ˆ t . White dashed line represents the limit between the UDV data and model regression. b) Time
evolution of the dimensionless streamwise velocity for different r: 0.0 (+), 0.1( � ), 0.2( � ), 0.3( � ), 0.4( �), 0.5( � ), 0.6( � ), 0.7( � ), 0.8( � ), 0.9 ( � ) and 1.0 ( ♦).
Fig. 6. Average streamwise velocity profiles for regions II ( �) and IV ( �
) corre-
sponding to the experiment shown in Fig. 5 . The solid line shows the fitted pro-
file obtained from the flow solution of a Herschel–Bulkley fluid. The Newtonian
Poiseuille profiles have also been indicated considering: (dashed line) and (dotted
line).
t
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p
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e
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o
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f
t
a
t
0
B
v
by an immiscible Newtonian fluid in a horizontal 2D channel, ana-
lyzed using lattice Boltzmann simulations.
Fig. 7. a) Snapshots of the concentration field for a 385 mm long section of the pipe
obtained for experimental set D given descending mean flow velocities: (a.1) ˆ V 0 = 47 . 4 m
and M = 811 ); (a.3) ˆ V 0 = 14 . 3 mm/s ( Re = 0 . 16 , B = 0 . 85 and M = 982 ); (a.4) ˆ V 0 = 4 . 4 m
concentration C . b) Depth-averaged concentration C y values with streamwise distance, x
(dotted line), a.3 (dashed line) and a.4 (dotted-dashed line).
The residual layer variation can also be identified from the spa-
iotemporal plot, as presented in Fig. 8 for different velocities be-
onging to experimental set C. The front velocity variation is very
rominent in Fig. 8 d, indicated by the non-linear boundary be-
ween the two (dark and light) fluid regions. The vertical traces
n the lighter region of Figs. 8 a–d correspond to the residual lay-
rs of viscoplastic fluid that remain static over long times after the
assage of the front. This also confirms that there is no wall slip
n our experiments. We note that there are considerable apparent
ifferences in the spatial distribution and amplitude of the residual
ayers.
The spatiotemporal diagram enables one to identify the posi-
ion and speed of the front at every time step. Therefore, instead
f showing the front velocity versus time, we can plot the front
elocity variation against the position of the front, i.e. ˆ V f ( x ) is the
ront velocity at the time when the front is at ˆ x . Figs. 9 a and b plot
he variation of the dimensionless front velocity V f (x ) =
ˆ V f ( x ) / V 0
nd the depth-averaged concentration C y field along the tube sec-
ion for two different experiments: low Reynolds number ( Re = . 02 , B = 1 . 51 , M = 1791 ) and high Reynolds number ( Re = 1 . 85 ,
= 0 . 3 , M = 378 ) respectively. Fig. 9 a shows a large amplitude
ariation of both front velocity and depth-averaged concentration
located 2286 mm downstream of the gate valve after the passage of the front
m/s ( Re = 1 . 20 , B = 0 . 47 and M = 532 ); (a.2) ˆ V 0 = 20 . 7 mm/s ( Re = 0 . 30 , B = 0 . 71
m/s ( Re = 0 . 02 , B = 1 . 51 and M = 1791 ). The bottom image is a colourbar of the
, after the front passage for the same mean velocity from a: a.1 (solid line), a.2
G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103 97
Fig. 8. Spatiotemporal diagrams of the same experiments shown in Fig. 7 .
Fig. 9. Variation of the mean depth-averaged concentration C y ( �) and front velocity V f ( �
) with the streamwise distance, ˆ x , for a) Re = 0.02, B = 1.51 and M = 1791 and
b)Re = 1.85, B = 0.3 and M = 378. Averages of V f and C y (dashed line) and the intervals of 95% confidence level (dotted lines) are also added to the figures.
w
o
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hereas Fig. 9 b shows a comparatively lower amplitude variation
f these variables. Note that, as suggested in these figures, the
epth-averaged concentration is dynamically coupled to the front
elocity, independently of Reynolds number, meaning that when
he front velocity increases, the local depth-averaged concentration
ecreases and vice versa.
From mass conservation we can define a parameter Q as fol-
ows, to measure the bulk experimental data quality
¯ =
ˆ V f ˆ d 2 C
ˆ V 0 D
2 = V f d
2 C = V f C , (10)
here, d C =
ˆ d C / D is the averaged dimensionless finger diameter of
uid 1, defined from C , which is the mean of the depth-averaged
oncentration C y . Also note the mean front velocity value, V f , is
sed in defining Q in (10) . For Q ≈ 1 , mass conservation is closely
espected. Slight deviation from this value ( Q = 1 or V f C = 1 ) may
appen due to various reasons namely image processing errors af-
ecting C calculation, V f approximation based on numerical deriva-
ives and inaccuracies due to the snapshots sampling frequency.
oreover, Q = 1 can be due to the fact that the residual layers
ight not be perfectly static with part of it being washed away
98 G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103
Fig. 10. a) Mean quality Q and uncertainty, versus the Reynolds number Re , for the whole range of experiments; different markers represent experimental batch A( �), B( �
),
C( ◦), D( � ), E( � ), F( � ), G( �), H( � ) and I( � ); overall quality average (solid line) and intervals of 95% of confidence level (dotted lines). b) Change in the quality uncertainty with
the Reynolds number ( �). The lines in b indicate: power law curve fitted to the data as δQ/ Q = 0 . 065 Re −0 . 4 (solid line); transition between the smooth and wavy regimes at
δQ/ Q = 0 . 067 (dotted line), transition between the wavy and corrugated regimes at δQ/ Q = 0 . 13 (dashed line).
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from the surface throughout the experiment, which consequently
affects mass conservation. Fig. 10 a shows the plot of quality Q ,
versus Reynolds number Re , suggesting an average of 0.86 over the
entire range of experiments. From (10) , we can calculate an uncer-
tainty of Q based on the standard deviation of the depth-averaged
concentration C y , denoted δC , and the standard deviation of the
spatially varying front velocity V f ( x ), denoted δV f . The uncertainty
of Q is defined as:
δQ
Q
=
√ (δC
C
)2
+
(δV f
V f
)2
(11)
where Q = V f C with C and V f representing the mean value of the
depth-averaged concentration and front velocity respectively.
Upon detailed analysis of the spatiotemporal diagrams of the
experiments, we have been able to classify isodense viscoplastic
displacement flows in three different categories, and match the ob-
served behaviours to ranges of the uncertainty of Q .
by a linear boundary between the two fluids in the spatiotem-
poral diagram (e.g. Fig. 8 a). These experiments were found for
δQ/ Q < 0 . 067 , as shown in Fig. 10 b. • Wavy : medium amplitude residual layer variations character-
ized by linear boundary between the two fluids in the spa-
tiotemporal diagram and clearly visible vertical stripes after the
passage of the front (e.g. Figs. 8 b and c). These flows were
found for 0 . 067 < δQ/ Q < 0 . 13 , as shown in Fig. 10 b. • Corrugated : large amplitude residual layer variation. In this cat-
egory, apart from observing well-defined vertical stripes after
the front has passed in the spatiotemporal diagram, there also
exists a non-linear boundary between the two fluids, due to
observable variations in the front velocity (e.g. Fig. 8 d). These
flows were found to have δQ/ Q > 0 . 13 ; see Fig. 10 b.
Fig. 10 b clearly shows that the quality uncertainty, δQ/ Q , which
combines the variability in both local concentration and front ve-
locity, corresponds visually to increased unevenness of the resid-
ual layers and this measure increases progressively from smooth
to corrugated flows.
The different flow regimes cannot be identified directly from
the UDV data, as they correspond to a spatial variability while the
velocity measurements are obtained at a specific location (UDV
robe). Figs. 5 and 6 earlier presented typical examples of UDV
ata for the smooth regime, where the residual layer is uniform
long the tube (see Fig. 5 a). In wavy and corrugated regimes, since
he residual layer thickness fluctuates along the pipe, the UDV data
an be affected as well. In fact, depending on the thickness of the
iscoplastic residual layer around the pipe at the UDV probe lo-
ation, the (displacing fluid) velocity can be shifted away from the
ipe centre. Fig. 11 illustrates this effect for two different sets of ex-
eriments classified as wavy . Note that there is some noise in the
ontours shown in Fig. 11 , which can be associated with the UDV
ignals and is particularly higher when the flow is stopped. Note
hat we have also centralized UDV contours for wavy and corru-
ated regimes, i.e. similar to the smooth regime shown in Fig. 5 a.
In a few corrugated flow regime experiments, we were able to
dentify more then one displacing fluid finger, as shown in Fig. 12 .
his phenomenon could only be detected by the UDV system, and
ay be associated to instabilities of the front, perhaps similar
o the tip-splitting mechanisms of [19] , which are not within the
cope of this study.
.3. Flow regime maps
We now consider how the main flow regimes defined in the
revious section ( smooth, wavy and corrugated ), correspond to vari-
tions in the dimensionless groups of the problem. Figs. 13 a and
show the flow regime classification plotted against dimension-
ess ( Re − B ) and ( Re − M) respectively. With the aid of Fig. 13 it
s possible to identify two transitions: smooth-wavy and wavy-
orrugated. The first transition occurs for Re ≈ 1 and the second
or Re ≈ 0.20.
Fig. 13 suggests that the Reynolds number controls the level of
he residual layer variation in viscoplastic displacement flow, em-
hasizing the importance of the inertial stress with respect to the
haracteristic viscous stresses of the displaced fluid. When inertial
tresses become dominant, the residual layer is uniform (smooth
egime) and fluid 1 flows smoothly in the center of the pipe. On
he other hand, as inertial stress weakens, the residual layer am-
litude variation increases, and the transition to the corrugated
egime occurs. The effects observed are generally in agreement
ith the observations of [11] , but are also counter-intuitive as nor-
ally we associate instability (unevenness) with increasing inertia.
G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103 99
Fig. 11. Contour of streamwise velocity obtained from UDV for experimental set G a) Re = 1 . 01 ; B = 0 . 4 and M = 747 , b) Re = 0 . 93 ; B = 0 . 4 and M = 768 and for experimental
set H c) Re = 0 . 47 ; B = 0 . 68 and M = 1002 , d) Re = 1 . 57 ; B = 0 . 50 and M = 693 .
Fig. 12. Example of isodense displacement with two fingers ( Re = 0 . 10 ; B = 1 . 14 and M = 1965 ): a) velocity contour, b)snapshots of the flow at ˆ t = 53 . 0 , 57 . 0 , 61 . 0 , 65 . 0 ,
and 69.0 s after opening the gate valve. The flow is from left to right as indicated by the arrow. The field of view is 583 mm by 19 mm, taken 2004 mm after the gate
valve. The bottom image is a colourbar of the concentration C .
e
n
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l
ε
The Reynolds number also influences the front velocity and the
ffective diameter of fluid 1 as shown in Fig. 14 . As the Reynolds
umber decreases, the front velocity V f increases ( Fig. 14 a) and the
quivalent mean diameter of fluid 1, d c , decreases ( Fig. 14 b). The
ffects observed are in qualitative agreement with the findings of
he numerical study of [15] .
We can use the front velocity ( V f ( x )) and/or the mean concen-
ration ( C y (x ) ) variation to calculate the roughness of the residual
ayer, given by:
i =
δd i
d i + δd i , i = f, C (12)
100 G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103
) plotted in dimensionless planes of a) B and Re, b) M and Re.
Fig. 14. a) Dimensionless front velocity, V f , as a function of the Reynolds number and power law data regression V f = 1 . 6 Re −0 . 03 (dashed line). b) Fluid 1 mean diameter,
d C =
√
C . Different markers represent experimental batch A( �), B( �
value of d i . If the front velocity is considered, i = f, and if C y (x ) is
considered, then i = C. Fig. 15 a shows how the roughness calcu-
lated from front velocity ( εf ) relates to the one calculated from the
mean concentration ( εC ). Since we have two data sets of rough-
ness coming from different image processing algorithms and both
are correlated, instead of using a specific set, an average roughness
value has been considered, defined as εeq = (εC + ε f ) / 2 . It can be
seen from Fig. 15 a that a large number of our flows locate within
the large roughness regime of Huang et al. [31] , (0.03 < ε < 0.33)
governed by completely different dynamics. They showed that the
flow in a duct with the hydraulic Reynolds number, Re h , and rela-
tive roughness, ε, can be laminar, transitional and/or turbulent . The
threshold Reynolds numbers were found to be
Re L = −6340 . 3 ε + 2304 . 4 , (13)
from laminar to transitional flow and
Re T = −3542 ln ε − 3739 . 6 , (14)
from transitional to turbulent flow. In order to determine which
of the laminar, transitional and turbulent regimes the Newtonian
fluid is flowing at, besides the equivalent roughness calculated in
Fig. 15 a, we need to compute the hydraulic Reynolds number for
region IV defined as
Re h =
ˆ ρ ˆ V f ˆ d c
ˆ μ. (15)
he results are presented in Fig. 15 b in the plane of Re h and εeq
ver our full range of experiments. We can see that most of the
xperiments fall in the laminar regime with a few locating in the
ransitional regime. Note that similar findings were witnessed for
on-isodense flows [12] . Also note that interestingly, the hydraulic
eynolds number seems to be inversely proportional to the equiva-
ent roughness ( Fig. 15 b), which is in line with our previous obser-
ation on the decrease of the interface modulations and variations
ith Re ( Fig. 10 b). The dotted lines corresponding to Re h = 13 /εeq ,
e h = 26 /εeq and Re h = 48 /εeq have been added to Fig. 15 b as eye
uide. Also note that it seems as if surface roughness values in iso-
ense displacement flows are overall lower than the ones in buoy-
nt flows (0 < ε < 0.25) [12] .
In order to check the validity of the results shown in Fig. 15 b on
redicting the flow of the displacing layer to be in laminar regime,
e have looked into the velocity profiles for a few flow examples
arked by black squares. The mean velocity profile in region IV
or these flows belonging to the experimental sets G and H are
resented in Figs. 16 (a),(b) and 16 (c),(d) respectively. The velocity
rofiles overall suggest an approximately parabolic shape which is
ypical of laminar flows. The corresponding curve-fitted Poiseuille
rofiles based on an average dimensionless displacing fluid diame-
er, (d f + d c ) / 2 , have been added to each case for comparison. Of
ourse we do not expect the velocity in this flow to be precisely
arabolic. There is some asymmetry evident in case (a) and (d),
G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103 101
Fig. 15. a) Correlation between the εC and ε f (solid symbols), εeq (hollow symbols), εC = ε f (solid line). b) Flow dynamics classification of fluid 1 in region IV (Laminar,
Transitional and Turbulent) in the plane of Re h and εeq . Re L = −6340 . 3 εeq + 2304 . 4 (thin solid line), Re T = −3542 ln εeq − 3739 . 6 (thick solid line), Re h = 13 /εeq (- - -), Re h =
26 /εeq (.-.-.-), Re h = 48 /εeq (.... ). Different markers represent experimental batch A( �), B( �
), C( ◦), D( � ), E( � ), F( � ), G( �), H( � ) and I( � ). The UDV profiles for the experiments
marked by black squares are given in Fig. 16 .
Fig. 16. Average velocity profile at region IV for a) experimental set G and Re = 1 . 01 , B = 0 . 4 , M = 747 , b) experimental set G and Re = 0 . 93 , B = 0 . 4 , M = 768 , c) experi-
mental set H and Re = 1 . 57 , B = 0 . 50 , M = 693 and d) experimental set H and Re = 0 . 47 , B = 0 . 68 , M = 1002 . The solid lines show the corresponding Poiseuille flow profiles
obtained from curve fitting of the data based an average dimensionless displacing fluid diameter, (d f + d c ) / 2 .
w
n
i
U
c
I
t
τ
w
w
s
τ
w
t
f
n
e
T
w
hich can be related to the skew in the UDV data [8] . There is
o clear unsteadiness and/or flatness appearing in the profiles sim-
lar to those of transitional and turbulent flows respectively. Some
DV examples of such flows are given in [12] for buoyant flows for
omparison.
The force balance for the developed steady state flow in region
V ( 1 / 4 π ˆ d 2 c ˆ p = π ˆ d c L τi ) gives the shear stress at the interface be-
ween the two fluids [12] :
ˆ i =
f ˆ ρ ˆ V
2 f
8
, (16)
here f is the Darcy friction factor. Since the stress varies linearly
ith the tube radius, we have the following expression for wall
hear stress
ˆ w
= ˆ τi
ˆ D
ˆ d f = ˆ τi
√
V f , (17)
hich is always larger than the interfacial stress, ˆ τi . The Darcy fric-
ion factor in laminar flow of a Newtonian fluid is given as f =
64 Re h
or small roughness values ( εeq < 0.05). However for large rough-
esses (0.05 < εeq < 0.3) Huang et al. [31] proposed the following
xpression obtained experimentally
f =
(10210 ε2
eq − 529 . 66 εeq + 64
)/Re h . (18)
he dependency of the percentage scaled wall shear stress, ˆ τw
/ τy ,
ith the hydraulic Reynolds number, Re , has been plotted in
h
102 G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103
Fig. 17. Dependency of the scaled wall shear stress in region IV with the hydraulic
Reynolds number, Re h , over the whole range of experiments. Corrugated flows are
marked with superimposed black dots. Different markers represent experimental
Fig. 17 . The figure shows that the wall stress in region IV is lower
than 5% of the yield stress of the viscoplastic fluid. Note that the
interfacial stress would be even lower than this by 1 / √
V f , accord-
ing to (17) .
This simple calculation shows that the Newtonian fluid flowing
inside a reduced diameter geometry (region IV) does not produce
a sufficiently large enough stress to yield the viscoplastic fluid at
the interface or wall, which explains the reason of having a static
residual layer after the front passage over long times. In other
words, the residual layer formation is purely related to the front
dynamics (region III).
4. Final remarks
In this work, we analyzed experimentally the isodense displace-
ment of a viscoplastic fluid by a Newtonian one in a horizontal
pipe. Velocity profiles and the fluids concentration field were ac-
quired for several pair of fluids through Ultrasonic Doppler Ve-
locimetry (UDV) and flow visualisation respectively. The displaced
fluids used were Carbopol solutions, while the Newtonian fluids
were salt or glycerol-water solutions. The major flow pattern is
characterized by a central displacement, with a residual layer of
the displaced fluid left close to the tube wall. Moreover, three dis-
tinct flow regimes could be identified within the central flows,
namely corrugated, wavy and smooth , depending on the level of
the residual layer variation along the pipe. The transition between
these flow regimes is found to be a function of the Reynolds num-
ber defined as the ratio of the inertial stress to the characteris-
tic stress of the viscoplastic fluid. In particular, Re = 0 . 2 and 1 for
wavy-corrugated and smooth-wavy flow transitions respectively.
The evaluation of the stress at the interface between the fluids
showed that it remains below the yield stress, therefore the for-
mation and modulation of the residual layer should be governed
by the dynamics happening at the frontal region of the displace-
ment.
cknowledgments
This research was supported financially by Petrobras S.A. and
razilian funding agencies (MCTI/CNPq, CAPES, FAPERJ and FINEP).
artial funding for the experimental programme is gratefully ac-
nowledged via CRD project 4 4 4985-12 (NSERC/Schlumberger).
he support of the University of Houston is also conceded. X. Dong
nd A. Gosselin are thanked for their help in running the experi-
ents. We also thank the reviewers for their helpful comments.
eferences
[1] R. Venkatesan , N.R. Nagarajan , K. Paso , Y.B. Yi , A.M. Sastry , H.S. Fogler ,
The strength of paraffin gels formed under static and flow conditions,Chem. Eng. Sc. 60 (13) (2005) 3587–3598 .
[2] I. Frigaard , G. Vinay , A. Wachs , Compressible displacement of waxy crude oilsin long pipeline startup flows, J. non-Newt. Fluid Mech. 147 (1) (2007) 45–64 .
[3] I. Dapra , G. Scarpi , Start-up flow of a Bingham fluid in a pipe, Meccanica 40
(1) (2005) 49–63 . [4] G. Vinay , Modélisation du redemarrage des écoulements de bruts parafiniques
dans les conduites pétrolieres, Ph.D. thesis, These de l’Ecole des Mines de Paris,Paris, France, 2005 .
[5] G. Vinay , A. Wachs , J.F. Agassant , Numerical simulation of weakly com-pressible Bingham flows: the restart of pipeline flows of waxy crude oils,
J. non-Newt. Fluid Mech. 136 (2) (2006) 93–105 .
[6] G. Vinay , A. Wachs , I. Frigaard , Start-up transients and efficient computationof isothermal waxy crude oil flows, J. non-Newt. Fluid Mech. 143 (2) (2007)
141–156 . [7] A. Wachs , G. Vinay , I. Frigaard , A 1.5 d numerical model for the start up of
weakly compressible flow of a viscoplastic and thixotropic fluid in pipelines,J. non-Newt. Fluid Mech. 159 (1) (2009) 81–94 .
[8] S.M. Taghavi , K. Alba , M. Moyers-Gonzalez , I.A. Frigaard , Incomplete fluid-fluiddisplacement of yield stress fluids in near-horizontal pipes: experiments and
theory, J. non-Newt. Fluid Mech. 167–168 (2012) 59–74 .
[9] K. Alba , S.M. Taghavi , J.R. de Bruyn , I.A. Frigaard , Incomplete flu-id–fluid displacement of yield-stress fluids. part 2: highly inclined pipes,
J. Non-Newt. Fluid Mech. 201 (2013) 80–93 . [10] M. Allouche , I.A. Frigaard , G. Sona , Static wall layers in the displacement of
two visco-plastic fluids in a plane channel, J. Fluid Mech. 424 (20 0 0) 243–277 .[11] C. Gabard , J.-P. Hulin , Miscible displacements of non-Newtonian fluids in a ver-
tical tube, Eur. Phys. J. E 11 (2003) 231–241 .
[12] K. Alba , I.A. Frigaard , Dynamics of the removal of viscoplastic fluids from in-clined pipes, J. non-Newt. Fluid Mech. 229 (2016) 43–58 .
[13] P.A. Cole , K. Asteriadou , P.T. Robbins , E.G. Owen , G.A. Montague , P.J. Fryer , Com-parison of cleaning of toothpaste from surfaces and pilot scale pipework, Food
Bioprod. Process. 88 (4) (2010) 392–400 . [14] I. Palabiyik , B. Olunloyo , P.J. Fryer , P.T. Robbins , Flow regimes in the empty-
ing of pipes filled with a Herschel–Bulkley fluid, Chem. Eng. Res. Des. 92 (11)
(2014) 2201–2212 . [15] K. Wielage-Burchard , I.A. Frigaard , Static wall layers in plane channel displace-
ment flows, J. Non-Newt. Fluid Mech. 166 (2011) 245–261 . [16] K. Alba , S.M. Taghavi , I.A. Frigaard , A weighted residual method for two-layer
non-Newtonian channel flows: steady-state results and their stability, J. FluidMech. 731 (2013) 509–544 .
[17] M. Moyers-Gonzalez , K. Alba , S.M. Taghavi , I.A. Frigaard , A semi-analytical clo-
sure approximation for pipe flows of two Herschel-Bulkley fluids with a strat-ified interface, J. non-Newt. Fluid Mech. 193 (2013) 49–67 .
[19] Y. Dimakopoulos , J. Tsamopoulos , Transient displacement of a viscoplastic ma-terial by air in straight and suddenly constricted tubes, J. non-Newt. Fluid
Mech. 112 (2003) 43–75 .
[20] Y. Dimakopoulos , J. Tsamopoulos , Transient displacement of Newtonian andviscoplastic liquids by air in complex tubes, J. non-Newt. Fluid Mech. 142
(2007) 162–182 . [21] P.R. de Souza Mendes , E.S.S. Dutra , J.R.R. Siffert , M.F. Naccache , Gas displace-
ment of viscoplastic liquids in capillary tubes, J. non-Newt. Fluid Mech. 145(1) (2007) 30–40 .
22] D.A. de Sousa , E.J. Soares , R.S. de Queiroz , R.L. Thompson , Numerical investiga-
tion on gas-displacement of a shear-thinning liquid and a visco-plastic mate-rial in capillary tubes, J. non-Newt. Fluid Mech. 144 (2) (2007) 149–159 .
[23] R.L. Thompson , E.J. Soares , R.D.A. Bacchi , Further remarks on numerical inves-tigation on gas displacement of a shear-thinning liquid and a visco-plastic ma-
terial in capillary tubes, J. non-Newt. Fluid Mech. 165 (7) (2010) 448–452 . [24] Y. Dimakopoulos , J. Tsamopoulos , On the gas-penetration in straight tubes
completely filled with a viscoelastic fluid, J. non-Newt. Fluid Mech. 117 (2)(2004) 117–139 .
in a plane channel with interfacial tension effects, Chem. Eng. Sci. 91 (2013)54–64 .
26] P.A.P. Swain , G. Karapetsas , O.K. Matar , K.C. Sahu , Numerical simulation ofpressure-driven displacement of a viscoplastic material by a Newtonian fluid
using the lattice Boltzmann method, Eur. J. Mech. B-Fluid 49 (2015) 197–207 .
G.V.L. Moisés et al. / Journal of Non-Newtonian Fluid Mechanics 236 (2016) 91–103 103
[
[
[
[27] S.M. Taghavi , K. Alba , T. Seon , K. Wielage-Burchard , D.M. Martinez , I.A. Frigaard ,Miscible displacement flows in near-horizontal ducts at low Atwood number,
J. Fluid Mech. 696 (2012) 175–214 . 28] K. Alba, Displacement flow of complex fluids in an inclined duct. (2013).
29] P. Coussot , Q.D. Nguyen , H.T. Huynh , D. Bonn , Avalanche behavior in yieldstress fluids, Phys. Rev. Lett. 88 (17) (2002) . 175501(1)–(4)
30] K. Alba , S.M. Taghavi , I.A. Frigaard , Miscible density-unstable displacementflows in inclined tube, Phys. Fluids 25 (2013) . 067101(1)–(21)
[31] K. Huang , J.W. Wan , C.X. Chen , Y.Q. Li , D.F. Mao , M.Y. Zhang , Exper-imental investigation on friction factor in pipes with large roughness,